Startseite Mathematik Asymptotic analysis of Leray solution for the incompressible NSE with damping
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Asymptotic analysis of Leray solution for the incompressible NSE with damping

  • Mongi Blel EMAIL logo und Jamel Benameur
Veröffentlicht/Copyright: 9. August 2024
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Demonstratio Mathematica
Aus der Zeitschrift Demonstratio Mathematica Band 57 Heft 1

Abstract

In 2008, Cai and Jiu showed that the Cauchy problem of the Navier-Stokes equations, with damping α ∣ u ∣ β − 1 u for α > 0 and β ≥ 1 has global weak solutions in L 2 ( R 3 ) . In this article, we study the uniqueness and the continuity in L 2 ( R 3 ) of this global weak solution. We also prove the large time decay for this global solution for β ≥ 10 3 .

Keywords: Navier-Stokes equations; Friedrich method; global weak solution
MSC 2010: Primary 35-XX; 35Q30; 76D05; 76N10

1 Introduction

Consider the three-dimensional Navier-Stokes equations (NSEs) with nonlinear damping

(NSD) ∂ t u − ν Δ u + u . ∇ u + α ∣ u ∣ β − 1 u = − ∇ p in R + × R 3 div u = 0 in R + × R 3 u ( 0 , x ) = u 0 ( x ) in R 3 α > 0 , β > 1 ,

where u = u ( t , x ) = ( u 1 , u 2 , u 3 ) and p = p ( t , x ) are, respectively, the unknown velocity and the pressure of the fluid at ( t , x ) ∈ R + × R 3 , ν represents the viscosity of the fluid, and u 0 = ( u 1 0 ( x ) , u 2 0 ( x ) , u 3 0 ( x ) ) represents the initial given velocity.

The NSEs govern the motion of incompressible fluids. The damping is from the resistance to the motion of the flow and describes various physical situations (see [1] and references therein).

The global existence of a weak solution to the initial value problem of the classical incompressible Navier-Stokes was proved by Leray and Hopf (see [2,3]) long before. The uniqueness remains an open problem for the dimensions d ≥ 3 .

The case of polynomial damping α ∣ u ∣ β − 1 u was studied in 2005 by Zhou [4] for α = 1 . He proved the existence of global strong solution for β ≥ 3 .

In 2008, Cai and Jiu [5] proved the existence of global weak solution for β ≥ 1 , the existence of global strong solution for β ≥ 7 2 , and the uniqueness of the strong solution for 7 2 ≤ β ≤ 5 . They employ the Galerkin approximations to construct the global solution of the system (NSD).

In 2011, Jia et al. [6] studied the L 2 decay of the weak solutions with β ≥ 10 3 and α = 1 .

For β > 3 and u 0 in a suitable space, Zhang et al. [7] proved the existence of global strong solution and they prove the uniqueness if 3 < β ≤ 5 . They also stated the existence and uniqueness of the strong solution to the system (NSD) for β > 3 , α > 0 , and α ≥ 1 2 if β = 3 .

In 2017, Liu and Gao [8] established the L 2 decay of the weak solutions for β > 2 and α > 0 . They also stated the asymptotic stability of the solution if β > 3 and α > 0 and if β = 3 for α ≥ 1 2 .

In this article, we study the uniqueness, continuity, and the large time decay of the global solution of the system (NSD). We use the Friedrich method to prove the continuity and the uniqueness of such solution for β > 3 . We also study the large time decay for β ≥ 10 3 . Precisely, our main result is the following:

Theorem 1.1

Let β > 3 and u 0 ∈ L 2 ( R 3 ) a divergence-free vector field, then there is a unique global solution of ( N S D ) : u ∈ C b ( R + , L 2 ( R 3 ) ) ∩ L 2 ( R + , H ˙ 1 ( R 3 ) ) ∩ L β + 1 ( R + , L β + 1 ( R 3 ) ) . Moreover, for all t ≥ 0

(1.1) ‖ u ( t ) ‖ L 2 2 + 2 ∫ 0 t ‖ ∇ u ( s ) ‖ L 2 2 d s + 2 α ∫ 0 t ‖ u ( s ) ∣ L β + 1 β + 1 d s ≤ ‖ u 0 ‖ L 2 2 .

Also, if β ≥ 10 3 , we have

(1.2) lim t → ∞ ‖ u ( t ) ‖ L 2 = 0 .

Remark 1.2

  1. In [6], the authors study the L 2 decay of weak solutions by developing the Fourier splitting method [9, 10] for α = 1 and β ≥ 10 3 . In this article, we use a different method called the Friedrich method to study the L 2 decay of the weak solution. We prove the uniqueness, the continuity of the global solution in L 2 ( R 3 ) , and the asymptotic behavior (1.2). This inequality is proved in [5].

  2. The case of fractional order NSEs with damping can be studied based on the previous work [11,12].

2 Preliminary results

In the following, we recall some preliminary results. The reader is referred to [12–14] for more details of the proofs.

Proposition 2.1

  1. In a Hilbert space H, the unit ball is weakly compact, i.e., if ( x n ) n is a sequence in the unit ball of H, then there is a subsequence ( x φ ( n ) ) n such that

    lim n → + ∞ ( x φ ( n ) ∣ y ) = ( x ∣ y ) , ∀ y ∈ H .

  2. If x ∈ H is a weak limit of a bounded sequence ( x n ) n in H , then

    ‖ x ‖ ≤ liminf n → ∞ ‖ x n ‖ .

  3. If x ∈ H is a weak limit of a bounded sequence ( x n ) n in H , and limsup n → ∞ ‖ x n ‖ ≤ ‖ x ‖ , then lim n → ∞ ‖ x n − x ‖ = 0 .

Lemma 2.2

Let s 1 and s 2 be two real numbers and N ∈ N .

  • If s 1 < N 2 and s 1 + s 2 > 0 , there exists a constant C 1 = C 1 ( N , s 1 , s 2 ) , such that if f , g ∈ H ˙ s 1 ( R N ) ∩ H ˙ s 2 ( R N ) , then f . g ∈ H ˙ s 1 + s 2 − N 2 ( R N ) and

    ‖ f g ‖ H ˙ s 1 + s 2 − N 2 ≤ C 1 ( ‖ f ‖ H ˙ s 1 ‖ g ‖ H ˙ s 2 + ‖ f ‖ H ˙ s 2 ‖ g ‖ H ˙ s 1 ) .

  • If s 1 , s 2 < N 2 and s 1 + s 2 > 0 , there exists a constant C 2 = C 2 ( N , s 1 , s 2 ) such that if f ∈ H ˙ s 1 ( R N ) and g ∈ H ˙ s 2 ( R N ) , then f . g ∈ H ˙ s 1 + s 2 − N 2 ( R N ) and

    ‖ f g ‖ H ˙ s 1 + s 2 − N 2 ≤ C 2 ‖ f ‖ H ˙ s 1 ‖ g ‖ H ˙ s 2 .

Lemma 2.3

Let r > 0 , then, for all x , y ∈ R N , we have

(2.1) ⟨ ∣ x ∣ r x − ∣ y ∣ r y , x − y ⟩ ≥ 1 2 ( ∣ x ∣ r + ∣ y ∣ r ) ∣ x − y ∣ 2 .

The Leray projector P : ( L 2 ( R 3 ) ) 3 → ( L σ 2 ( R 3 ) ) 3 is defined by

ℱ ( P f ) = f ^ ( ξ ) − f ^ ( ξ ) . ξ ∣ ξ ∣ ξ ∣ ξ ∣ = M ( ξ ) f ^ ( ξ ) ,

where M ( ξ ) is the matrix δ k , ℓ − ξ k ξ ℓ ∣ ξ ∣ 2 1 ≤ k , ℓ ≤ 3 and L σ 2 ( R 3 ) represents the space of divergence-free vector fields in L 2 ( R 3 ) . In particular, if u is in the Schwartz space ( S ( R 3 ) ) 3 , the Leray projector P is given by

P ( u ) k ( x ) = 1 ( 2 π ) 3 2 ∫ R 3 δ k j − ξ k ξ j ∣ ξ ∣ 2 u ^ j ( ξ ) e i ξ ⋅ x d ξ .

For R > 0 , the Friedrich operator J R is defined on L 2 ( R 3 ) by J R ( D ) f = ℱ − 1 ( χ B R f ^ ) and the operator A R ( D ) is defined by

A R ( D ) u = P J R ( D ) u = ℱ − 1 ( M ( ξ ) χ B R ( ξ ) u ^ ) ,

with B R the ball of center 0 and radius R > 0 .

The following result is a generalization of Proposition 3.1 in [15]. The proof is similar to that of Proposition 2.4 in [12].

Proposition 2.4

Let ν 1 , ν 2 , ν 3 ∈ [ 0 , ∞ ) , r 1 , r 2 , r 3 ∈ ( 0 , ∞ ) , and f 0 ∈ L σ 2 ( R 3 ) .

For n ∈ N , let F n : R + × R 3 → R 3 be a measurable function in C 1 ( R + , L 2 ( R 3 ) ) such that A n ( D ) F n = F n , F n ( 0 , x ) = A n ( D ) f 0 ( x ) , and

∂ t F n + ∑ k = 1 3 ν k ∣ D k ∣ 2 r k F n + A n ( D ) div ( F n ⊗ F n ) + A n ( D ) h ( ∣ F n ∣ ) F n = 0 .

‖ F n ( t ) ‖ L 2 2 + 2 ∑ k = 1 3 ν k ∫ 0 t ‖ ∣ D k ∣ r k F n ( s ) ‖ L 2 2 d s + 2 ∫ 0 t ‖ h ( ∣ F n ( s ) ∣ ) ∣ F n ( s ) ∣ 2 ‖ L 1 d s ≤ ‖ f 0 ‖ L 2 2 ,

where h ( z ) = α z β − 1 , with α > 0 and β > 3 . Then: for every ε > 0 there is δ = δ ( ε , α , β , ν 1 , ν 2 , ν 3 , r 1 , r 2 , r 3 , ‖ f 0 ‖ L 2 ) > 0 such that for all t 1 , t 2 ∈ R + , we have

( ∣ t 2 − t 1 ∣ < δ ⇒ ‖ F n ( t 2 ) − F n ( t 1 ) ‖ H − s 0 < ε ) , ∀ n ∈ N ,

with s 0 ≥ max ( 3 , 2 r 1 , 2 r 2 , 2 r 3 ) .

3 Proof of Theorem 1.1

3.1 Existence of solution

Consider the approximate system:

( NSD n ) ∂ t u − Δ J n u + J n ( J n u . ∇ J n u ) + α J n [ ∣ J n u ∣ β − 1 J n u ] = − ∇ p n in R + × R 3 p n = ( − Δ ) − 1 ( div J n ( J n u . ∇ J n u ) + α div J n [ ∣ J n u ∣ β − 1 J n u ] ) div u = 0 in R + × R 3 u ( 0 , x ) = J n u 0 ( x ) in R 3 .

  • By the Cauchy-Lipschitz theorem, there exists a unique solution u n ∈ C 1 ( R + , L σ 2 ( R 3 ) ) of the system ( NSD n ) such that J n u n = u n and

    (3.1) ‖ u n ( t ) ‖ L 2 2 + 2 ∫ 0 t ‖ ∇ u n ( s ) ‖ L 2 2 d s + 2 α ∫ 0 t ‖ u n ( s ) ‖ L β + 1 β + 1 d s ≤ ‖ u 0 ‖ L 2 2 .

  • The sequence ( u n ) n is bounded on L 2 ( R 3 ) and H 1 ( R 3 ) . Hence, by Proposition 2.4 and the interpolation method, the sequence ( u n ) n is equicontinuous on H − 1 ( R 3 ) .

  • For a strictly increasing sequence ( T k ) k such that lim k → + ∞ T k = ∞ , consider a sequence of functions ( δ k ) k in C 0 ∞ ( R 3 ) such that

    δ k ( x ) = 1 , for ∣ x ∣ ≤ k + 5 4 δ k ( x ) = 0 , for ∣ x ∣ ≥ k + 2 0 ≤ δ k ≤ 1 .

    Using the energy estimate (3.1), the equicontinuity of the sequence ( u n ) n on H − 1 ( R 3 ) and classical argument by combining Ascoli’s theorem and the Cantor diagonal process, there exists a subsequence ( u φ ( n ) ) n and u ∈ L ∞ ( R + , L 2 ( R 3 ) ) ∩ C ( R + , H − 3 ( R 3 ) ) such that for all k ∈ N ,

    (3.2) lim n → ∞ ‖ δ k ( u φ ( n ) − u ) ‖ L ∞ ( [ 0 , T k ] , H − 4 ) = 0 .

    In particular, the sequence ( u φ ( n ) ( t ) ) n is weakly convergent in L 2 ( R 3 ) to u ( t ) for all t ≥ 0 .

  • Using the same method as in [15], we obtain

    (3.3) ‖ u ( t ) ‖ L 2 2 + 2 ∫ 0 t ‖ ∇ u ( s ) ‖ L 2 2 d s + 2 α ∫ 0 t ‖ u ( s ) ‖ L β + 1 β + 1 d s ≤ ‖ u 0 ‖ L 2 2 ,

    for all t ≥ 0 , and u a solution of the system ( N S D ) .

3.2 Continuity of the solution in L 2

Using inequality (3.3), we obtain limsup t → 0 ‖ u ( t ) ‖ L 2 ≤ ‖ u 0 ‖ L 2 and by Proposition 2.1-(3), we have limsup t → 0 ‖ u ( t ) − u 0 ‖ L 2 = 0 . This ensures the continuity of the solution u at 0. To prove the continuity on R + , consider the functions v n , ε ( t ) = u φ ( n ) ( t + ε ) , p n , ε ( t ) = p φ ( n ) ( t + ε ) , for n ∈ N and ε > 0 . We have

∂ t u φ ( n ) − Δ u φ ( n ) + J φ ( n ) ( u φ ( n ) . ∇ u φ ( n ) ) + α J φ ( n ) ( ∣ u φ ( n ) ∣ β − 1 u φ ( n ) ) = − ∇ p φ ( n ) ∂ t v n , ε − Δ v n , ε + J φ ( n ) ( v n , ε . ∇ v n , ε ) + α J φ ( n ) ( ∣ v n , ε ∣ β − 1 v n , ε ) = − ∇ p n , ε .

The function w n , ε = u φ ( n ) − v n , ε verifies the following:

∂ t w n , ε − Δ w n , ε + α J φ ( n ) ( ∣ u φ ( n ) ∣ β − 1 u φ ( n ) − ∣ v φ ( n ) ∣ β − 1 v n , ε ) = − ∇ ( p φ ( n ) − p n , ε ) + J φ ( n ) ( w n , ε . ∇ w n , ε ) − J φ ( n ) ( w n , ε . ∇ u φ ( n ) ) − J φ ( n ) ( u φ ( n ) . ∇ w n , ε ) .

Taking the scalar product with w n , ε in L 2 ( R 3 ) and using that div w n , ε = 0 and ⟨ w n , ε . ∇ w n , ε , w n , ε ⟩ = 0 , we obtain

(3.4) 1 2 d d t ‖ w n , ε ( t ) ‖ L 2 2 + ‖ ∇ w n , ε ( t ) ‖ L 2 2 + α ⟨ J φ ( n ) ( ∣ u φ ( n ) ∣ β − 1 u φ ( n ) − ∣ v n , ε ∣ β − 1 v n , ε ) ; w n , ε ⟩ L 2 = − ⟨ J φ ( n ) ( w n , ε . ∇ u φ ( n ) ) ; w n , ε ⟩ L 2 .

By inequality (2.1), we have

⟨ J φ ( n ) ( ∣ u φ ( n ) ∣ β − 1 u φ ( n ) − ∣ v n , ε ∣ β − 1 v n , ε ) ; w n , ε ⟩ L 2 = ⟨ ∣ u φ ( n ) ∣ β − 1 u φ ( n ) − ∣ v n , ε ∣ β − 1 v n , ε ; J φ ( n ) w n , ε ⟩ L 2 = ⟨ ∣ u φ ( n ) ∣ β − 1 u φ ( n ) − ∣ v n , ε ∣ β − 1 v n , ε ; w n , ε ⟩ L 2 ≥ 1 2 ∫ R 3 ( ∣ u φ ( n ) ∣ β − 1 + ∣ v n , ε ∣ β − 1 ) ∣ w n , ε ∣ 2 ≥ 1 2 ∫ R 3 ∣ u φ ( n ) ∣ β − 1 ∣ w n , ε ∣ 2 ,

which implies

(3.5) α ⟨ J φ ( n ) ( ∣ u φ ( n ) ∣ β − 1 u φ ( n ) − ( ∣ v n , ε ∣ β − 1 v n , ε ) ) ; w n , ε ⟩ L 2 ≥ α 2 ∫ R 3 ∣ u φ ( n ) ∣ β − 1 ∣ w n , ε ∣ 2 .

Also, we have

∣ ⟨ J φ ( n ) ( w n , ε . ∇ u φ ( n ) ) ; w n , ε ⟩ L 2 ∣ ≤ ∫ R 3 ∣ w n , ε ∣ . ∣ u φ ( n ) ∣ . ∣ ∇ w n , ε ∣ ≤ 1 2 ∫ R 3 ∣ w n , ε ∣ 2 ∣ u φ ( n ) ∣ 2 + 1 2 ‖ ∇ w n , ε ‖ L 2 2 .

The convex inequality a b ≤ a p p + b q q ≤ a p + b q , with p = β − 1 2 , q = β − 1 β − 3 , a = ∣ w n , ε ∣ 2 ( α 2 ) 2 β − 1 , b = ( 2 α ) 2 β − 1 , yields

∣ ⟨ J φ ( n ) ( w n , ε . ∇ u φ ( n ) ) ; w n , ε ⟩ L 2 ∣ ≤ α 4 ∫ R 3 ∣ w n , ε ∣ β − 1 ∣ u φ ( n ) ∣ 2 + C α , β ‖ w n , ε ‖ L 2 2 + 1 2 ‖ ∇ w n , ε ‖ L 2 2 ,

where C α , β = 1 2 ( 2 α ) 2 β − 3 . Combining this inequality and inequalities (3.4), (3.5), and (3.6), we obtain

d d t ‖ w n , ε ‖ L 2 2 + ‖ ∇ w n , ε ‖ L 2 2 ≤ 2 C α , β ‖ w n , ε ‖ L 2 2 .

By the Gronwall lemma, we deduce

‖ w n , ε ( t ) ‖ L 2 ≤ ‖ w n , ε ( 0 ) ‖ L 2 e C α , β t

and

‖ u φ ( n ) ( t + ε ) − u φ ( n ) ( t ) ‖ L 2 ≤ ‖ u φ ( n ) ( ε ) − u φ ( n ) ( 0 ) ‖ L 2 e C α , β t .

For t 0 > 0 and ε ∈ ( 0 , t 0 ) , we have

‖ u φ ( n ) ( t 0 + ε ) − u φ ( n ) ( t 0 ) ‖ L 2 ≤ ‖ u φ ( n ) ( ε ) − u φ ( n ) ( 0 ) ‖ L 2 e C α , β t 0 .

‖ u φ ( n ) ( t 0 − ε ) − u φ ( n ) ( t 0 ) ‖ L 2 ≤ ‖ u φ ( n ) ( ε ) − u φ ( n ) ( 0 ) ‖ L 2 e C α , β t 0 .

So

‖ u φ ( n ) ( ε ) − u φ ( n ) ( 0 ) ‖ L 2 2 = ‖ J φ ( n ) u φ ( n ) ( ε ) − J φ ( n ) u φ ( n ) ( 0 ) ‖ L 2 2 = ‖ J φ ( n ) ( u φ ( n ) ( ε ) − u 0 ) ‖ L 2 2 ≤ ‖ u φ ( n ) ( ε ) − u 0 ‖ L 2 2 ≤ ‖ u φ ( n ) ( ε ) ‖ L 2 2 + ‖ u 0 ‖ L 2 2 − 2 R e ⟨ u φ ( n ) ( ε ) , u 0 ⟩ ≤ 2 ‖ u 0 ‖ L 2 2 − 2 R e ⟨ u φ ( n ) ( ε ) , u 0 ⟩ .

But lim n → + ∞ ⟨ u φ ( n ) ( ε ) , u 0 ⟩ = ⟨ u ( ε ) , u 0 ⟩ , hence

liminf n → ∞ ‖ u φ ( n ) ( ε ) − u φ ( n ) ( 0 ) ‖ L 2 2 ≤ 2 ‖ u 0 ‖ L 2 2 − 2 Re ⟨ u ( ε ) ; u 0 ⟩ L 2 .

Moreover, for all k , N ∈ N

‖ J N ( δ k . ( u φ ( n ) ( t 0 ± ε ) − u φ ( n ) ( t 0 ) ) ) ‖ L 2 2 ≤ ‖ δ k . ( u φ ( n ) ( t 0 ± ε ) − u φ ( n ) ( t 0 ) ) ‖ L 2 2 ≤ ‖ u φ ( n ) ( t 0 ± ε ) − u φ ( n ) ( t 0 ) ‖ L 2 2 .

Using (3.2) we obtain, for k big enough,

‖ J N ( δ k . ( u ( t 0 ± ε ) − u ( t 0 ) ) ) ‖ L 2 ≤ liminf n → ∞ ‖ u φ ( n ) ( t 0 ± ε ) − u φ ( n ) ( t 0 ) ‖ L 2 .

Then

‖ J N ( δ k . ( u ( t 0 ± ε ) − u ( t 0 ) ) ) ‖ L 2 2 ≤ 2 ( ‖ u 0 ‖ L 2 2 − Re ⟨ u ( ε ) ; u 0 ⟩ L 2 ) e 2 C α , β t 0 .

By applying the monotone convergence theorem in the order N and q , we obtain

‖ u ( t 0 ± ε ) − u ( t 0 ) ‖ L 2 2 ≤ 2 ( ‖ u 0 ‖ L 2 2 − R e ⟨ u ( ε ) ; u 0 ⟩ L 2 ) e 2 C α , β t 0 .

Using the continuity at 0, we deduce the continuity at t 0 by making ε → 0 .

3.3 Uniqueness

Let u and v be two solutions of ( N S D ) in the space

C b ( R + , L 2 ( R 3 ) ) ∩ L 2 ( R + , H ˙ 1 ( R 3 ) ) ∩ L β + 1 ( R + , L β + 1 ( R 3 ) ) .

The function w = u − v satisfies the following:

∂ t w − Δ w + α ( ∣ u ∣ β − 1 u − ∣ v ∣ β − 1 v ) = − ∇ ( p − p ˜ ) + w . ∇ w − w . ∇ u − u . ∇ w .

Taking the scalar product in L 2 with w , we obtain

1 2 d d t ‖ w ‖ L 2 2 + ‖ ∇ w ‖ L 2 2 + α ⟨ ( ∣ u ∣ β − 1 u − ∣ v ∣ β − 1 v ) ; w ⟩ L 2 = − ⟨ w . ∇ u ; w ⟩ L 2 .

By adapting the same method for the proof of the continuity of such solution in L 2 ( R 3 ) , with u , v , w instead of u φ ( n ) , v n , ε , w n , ε in order, we find

α ⟨ ( ∣ u ∣ β − 1 u − ∣ v ∣ β − 1 v ) ; w ⟩ L 2 ≥ α 2 ∫ R 3 ∣ u ∣ β − 1 ∣ w ∣ 2

and

∣ ⟨ w . ∇ u ; w ⟩ L 2 ∣ ≤ α 4 ∫ R 3 ∣ w ∣ 2 ∣ u ∣ 2 + C α , β ‖ w ‖ L 2 2 + 1 2 ‖ ∇ w ‖ L 2 2 .

Combining the above inequalities, we obtain the following energy estimate:

d d t ‖ w ( t ) ‖ L 2 2 + ‖ ∇ w ( t ) ‖ L 2 2 ≤ 2 C α , β ‖ w ( t ) ‖ L 2 2 .

By Gronwall lemma inequality, we obtain

‖ w ( t ) ‖ L 2 2 + ∫ 0 t ‖ ∇ w ‖ L 2 2 ≤ ‖ w 0 ‖ L 2 2 e 2 C α , β t .

As w 0 = 0 , then w = 0 and u = v , which implies the uniqueness.

3.4 Asymptotic study of the global solution

To prove the asymptotic behavior (1.2), we need some preliminary lemmas:

Lemma 3.1

If u is a global solution of ( N S D ) with β ≥ 10 3 , then u ∈ L β ( R + × R 3 ) .

Proof

Let E 1 = { ( t , x ) : ∣ u ( t , x ) ∣ ≤ 1 } , E 2 = { ( t , x ) : ∣ u ( t , x ) ∣ > 1 } , L 1 = ∫ E 1 ∣ u ( s , x ) ∣ β d x d s , and L 2 = ∫ E 2 ∣ u ( s , x ) ∣ β d x d s . We have

L 1 = ∫ E 1 ∣ u ( s , x ) ∣ β d x d s = ∫ E 1 ∣ u ( s , x ) ∣ β − 10 3 ∣ u ( s , x ) ∣ 10 3 d x d s ≤ ∫ 0 ∞ ‖ u ( s ) ‖ L 10 3 10 3 d s .

Using the Sobolev injection H ˙ 3 5 ( R 3 ) ↪ L 10 3 ( R 3 ) , we obtain

L 1 ≤ C ∫ 0 ∞ ‖ u ( s ) ‖ H ˙ 3 5 10 3 d s .

By interpolation inequality ‖ u ( s ) ‖ H ˙ 3 5 ≤ ‖ u ( s ) ‖ H ˙ 0 2 5 ‖ u ( s ) ‖ H ˙ 1 3 5 , we obtain

L 1 ≤ C ∫ 0 ∞ ‖ u ( s ) ‖ L 2 4 3 ‖ ∇ u ( s ) ‖ L 2 2 ≤ C ‖ u 0 ‖ L 2 4 3 ∫ 0 ∞ ‖ ∇ u ( s ) ‖ L 2 2 .

Moreover, for the term L 2 , we have

L 2 = ∫ E 2 ∣ u ( s , x ) ∣ β d x d s ≤ ∫ 0 ∞ ∫ R 3 ∣ u ( s , x ) ∣ β + 1 d x d s .

Hence

‖ u ‖ L β ( R + × R 3 ) ≤ C ‖ u 0 ‖ L 2 4 3 ∫ 0 ∞ ‖ ∇ u ( s ) ‖ L 2 2 d s + ∫ 0 ∞ ∫ R 3 ∣ u ( s , x ) ∣ β + 1 d x d s .

Therefore, u ∈ L β ( R + × R 3 ) .□

Lemma 3.2

If u is a global solution of ( N S D ) , with β ≥ 10 3 , then lim t → ∞ ‖ u ( t ) ‖ H − 2 = 0 .

Proof

For ε > 0 , using the energy inequality (1.1) and Lemma 3.1, there exists t 0 ≥ 0 such that

(3.6) ‖ ∇ u ‖ L 2 ( [ t 0 , ∞ ) × R 3 ) < ε 4

and

(3.7) ‖ u ‖ L β ( [ t 0 , ∞ ) × R 3 ) < ε 4 .

Now, consider the following system:

(NSD′) ∂ t v − ν Δ v + v . ∇ v + α ∣ v ∣ β − 1 v = − ∇ q in R + × R 3 div v = 0 in R + × R 3 v ( 0 , x ) = u ( t 0 , x ) in R 3 .

By the existence and uniqueness part, the system ( NSD ′ ) has a unique global solution v ∈ C b ( R + , L 2 ( R 3 ) ) ∩ L 2 ( R + , H ˙ 1 ( R 3 ) ) ∩ L β + 1 ( R + , L β + 1 ( R 3 ) ) such that v ( t 0 ) = u ( t 0 , x ) and q ( t ) = p ( t 0 + t ) . The energy estimate for this system is as follows:

‖ v ( t ) ‖ L 2 2 + 2 ∫ 0 t ‖ ∇ v ( s ) ‖ L 2 2 d s + 2 a ∫ 0 t ‖ v ( s ) ‖ L β + 1 β + 1 ≤ ‖ u ( t 0 ) ‖ L 2 2 ≤ ‖ u 0 ‖ L 2 2 .

By the Duhamel formula, we obtain

v ( t , x ) = e t Δ v 0 ( x ) + f ( t , x ) + g ( t , x ) ,

where

f ( t , x ) = − ∫ 0 t e ( t − s ) Δ P div ( v ⊗ v ) ( s , x ) d s

and

g ( t , x ) = − α ∫ 0 t e ( t − s ) Δ P div ∣ v ( s , x ) ∣ β − 1 v ( s , x ) d s .

By dominated convergence theorem, lim t → ∞ ‖ e t Δ v 0 ‖ L 2 = 0 and hence lim t → ∞ ‖ e t Δ v 0 ‖ H − 2 = 0 . Moreover,

‖ f ( t ) ‖ H − 2 2 ≤ ‖ f ( t ) ‖ H − 1 2 2 ≤ ‖ f ( t ) ‖ H ˙ − 1 2 2 ≤ ∫ R 3 ∣ ξ ∣ − 1 ∫ 0 t e − ( t − s ) ∣ ξ ∣ 2 ∣ ℱ div ( v ⊗ v ) ( s , ξ ) ∣ d s 2 d ξ ≤ ∫ R 3 ∣ ξ ∣ ∫ 0 t e − ( t − s ) ∣ ξ ∣ 2 ∣ ℱ ( v ⊗ v ) ( s , ξ ) ∣ d s 2 d ξ .

Since

∫ 0 t e − ( t − s ) ∣ ξ ∣ 2 ∣ ℱ ( v ⊗ v ) ( s , ξ ) ∣ d s 2 ≤ ∫ 0 t e − 2 ( t − s ) ∣ ξ ∣ 2 d s ∫ 0 t ∣ ℱ ( v ⊗ v ) ( s , ξ ) ∣ 2 d s ≤ ∣ ξ ∣ − 2 ∫ 0 t ∣ ℱ ( v ⊗ v ) ( s , ξ ) ∣ 2 d s ,

then

‖ f ( t ) ‖ H − 2 2 d t ≤ ∫ R 3 ∣ ξ ∣ − 1 ∫ 0 t ∣ ℱ ( v ⊗ v ) ( s , ξ ) ∣ 2 d s d ξ ≤ ∫ 0 t ∫ R 3 ∣ ξ ∣ − 1 ∣ ( v ⊗ v ) ( s , ξ ) ∣ 2 d ξ d s = ∫ 0 t ‖ ( v ⊗ v ) ( s ) ‖ H ˙ − 1 2 2 d s .

Using the product law in homogeneous Sobolev spaces, with s 1 = 0 , s 2 = 1 , we obtain

‖ f ( t ) ‖ H − 2 2 d t ≤ C ∫ 0 t ‖ v ( s ) ‖ L 2 2 ‖ ∇ v ( s ) ‖ L 2 2 d s .

Using inequalities (3.6) and (3.7), we obtain

‖ f ( t ) ‖ H − 2 2 d t ≤ C ‖ u 0 ‖ L 2 2 ∫ 0 t ‖ ∇ u ( t 0 + s ) ‖ L 2 2 d s ≤ C ‖ u 0 ‖ L 2 2 ∫ 0 ∞ ‖ ∇ u ( t 0 + s ) ‖ L 2 2 d s ≤ C ‖ u 0 ‖ L 2 2 ∫ t 0 ∞ ‖ ∇ u ( s ) ‖ L 2 2 d s ≤ C ‖ u 0 ‖ L 2 2 ε 2 9 ( C ‖ u 0 ‖ L 2 2 + 1 ) ,

which implies that ‖ f ( t ) ‖ H − 2 < ε 3 , ∀ t ≥ 0 .

For an estimation of ‖ g ( t ) ‖ H − 2 , using the injection L 1 ( R 3 ) ↪ H − s ( R 3 ) , ∀ s > 3 ∕ 2 , with s = 2 , we obtain

‖ g ( t ) ‖ H − 2 2 d t ≤ ∫ R 3 ( 1 + ∣ ξ ∣ 2 ) − 2 ∫ 0 t e − ( t − s ) ∣ ξ ∣ 2 ∣ ℱ ( ∣ v ∣ β − 1 v ) ( s , ξ ) ∣ d s 2 d ξ ≤ C ∫ 0 t ‖ ( ∣ v ∣ β − 1 v ) ( s , . ) ‖ L 1 ( R 3 ) d s 2 ≤ C ∫ 0 t ‖ ∣ v ( s , . ) ∣ β ‖ L 1 ( R 3 ) d s 2 ≤ C ‖ v ‖ L β ( R + × R 3 ) 2 ,

where C = ∫ R 3 ( 1 + ∣ ξ ∣ 2 ) − 2 d ξ .

Also, using inequality (3.7), we obtain

‖ g ( t ) ‖ H − 2 2 d t ≤ C ‖ ∣ u ( t 0 + . ) ‖ L β ( R + × R 3 ) 2 ≤ C ‖ u ‖ L β ( [ t 0 , ∞ ) × R 3 ) 2 ≤ C ε 2 9 C ,

which implies that ‖ g ( t ) ‖ H − 2 < ε 3 , ∀ t ≥ 0 . Combining the above inequalities, we obtain

□ lim t → ∞ ‖ u ( t ) ‖ H − 2 = 0 .

Lemma 3.3

If u is a global solution of ( N S D ) and β ≥ 10 3 , then lim t → ∞ ‖ u ( t ) ‖ L 2 = 0 .

Proof

Let w 1 = 1 ∣ D ∣ < 1 u = ℱ − 1 ( 1 ∣ ξ ∣ < 1 u ^ ) and w 2 = 1 ∣ D ∣ ≥ 1 u = ℱ − 1 ( 1 ∣ ξ ∣ ≥ 1 u ^ ) .

Using the second step, we obtain

‖ w 1 ( t ) ‖ L 2 = c 0 ‖ w 1 ( t ) ‖ H 0 ≤ 2 c 0 ‖ ∣ w 1 ( t ) ‖ H − 2 ≤ 2 ‖ ∣ u ( t ) ‖ H − 2 ,

which implies lim t → ∞ ‖ w 1 ( t ) ‖ L 2 = 0 .

For ε > 0 , there is t 1 > 0 such that ‖ w 1 ( t ) ‖ L 2 < ε 2 , ∀ t ≥ t 1 , we have

∫ t 1 ∞ ‖ w 2 ( t ) ‖ L 2 2 d t ≤ ∫ t 1 ∞ ‖ ∇ w 2 ( t ) ‖ L 2 2 d t ≤ ∫ t 1 ∞ ‖ ∇ u ( t ) ‖ L 2 2 d t < ∞ .

Since the map t ⟼ ‖ w 2 ( t ) ‖ L 2 is continuous, there exists t 2 ≥ t 1 such that ‖ w 2 ( t 2 ) ‖ L 2 < ε 2 . Hence

‖ u ( t 2 ) ‖ L 2 2 = ‖ w 1 ( t 2 ) ‖ L 2 2 + ‖ w 2 ( t 2 ) ‖ L 2 2 < ε 2 2 .

Using the following energy estimate

‖ u ( t ) ‖ L 2 2 + 2 ∫ t 2 t ‖ ∇ u ( s ) ‖ L 2 2 d s + 2 α ∫ t 2 t ‖ u ( s ) ‖ L β + 1 d s ≤ ‖ u ( t 2 ) ‖ L 2 2 , ∀ t ≥ t 2 ,

we obtain ‖ u ( t ) ‖ L 2 < ε , ∀ t ≥ t 2 , and the proof is completed.□

4 Conclusion

Cai and Jiu [5] consider the incompressible NSEs with damping (NSD) and they proved the existence of global weak solution for β ≥ 1 . The problem of uniqueness and asymptotic behavior of the solution is not addressed. In this article, we prove by a new and different method, the uniqueness and the continuity of the global solution in L 2 ( R 3 ) for β > 3 . We also study the asymptotic behavior for this global solution for β ≥ 10 3 .

Acknowledgement

This project was supported by Researchers Supporting Project RSPD2024R753, King Saud University, Riyadh, Saudi Arabia.

  1. Funding information: This research has received funding support from King Saud University (Project RSPD2024R753).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

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Received: 2023-05-18
Revised: 2023-10-29
Accepted: 2024-06-09
Published Online: 2024-08-09

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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Navier-Stokes equations; Friedrich method; global weak solution
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Artikel in diesem Heft

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
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Heruntergeladen am 4.2.2026 von https://www.degruyterbrill.com/document/doi/10.1515/dema-2024-0042/html
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